Supervisor: Dr Natasha Blitvic
Project description:
Which of the familiar sequences from combinatorics are moments of probability measures on the real line? For instance, n!, counting all permutations on n letters, is the nth moment of the rate 1 exponential distribution, and the Bell numbers, counting all set partitions, are moments of the rate 1 Poisson. However, the answers can be very hard to predict. For example, the number of trees on labeled nodes is a moment sequence, whereas the number of trees on unlabeled nodes is not.
Being able to enumerate discrete structures is a critical aspect of probability, computer science, physics, and many other areas, and representing a sequence as an integral against a positive measure opens up new tools for understanding its behavior and properties, particularly when it comes to difficult enumeration problems. This project will develop new techniques for obtaining these types of representations, focusing on several famous sequences for which this type of representation is conjectured to exist. This project will exploit the synergy between the work done at QMUL and Prof Alan Sokal’s group at UCL.
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